Selected methods of the determination of core inflation

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Publishing House of Wrocław University of Economics Wrocław 2015

Financial Investments and Insurance –

Global Trends and the Polish Market

PRACE NAUKOWE

Uniwersytetu Ekonomicznego we Wrocławiu

RESEARCH PAPERS

of Wrocław University of Economics

Nr

381 edited by

Krzysztof Jajuga

Wanda Ronka-Chmielowiec

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Copy-editing: Agnieszka Flasińska Layout: Barbara Łopusiewicz Proof-reading: Barbara Cibis Typesetting: Małgorzata Czupryńska Cover design: Beata Dębska

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6

Contents Adam Marszk: Stock markets in BRIC: development levels and

macroeco-nomic implications ... 250

Aleksander R. Mercik: Counterparty credit risk in derivatives ... 264 Josef Novotný: Possibilities for stock market investment using psychological

analysis ... 275

Krzysztof Piasecki: Discounting under impact of temporal risk aversion −

a case of discrete time ... 289

Aleksandra Pieloch-Babiarz: Dividend initiation as a signal of subsequent

earnings performance – Warsaw trading floor evidence ... 299

Radosław Pietrzyk, Paweł Rokita: On a concept of household financial plan

optimization model ... 314

Agnieszka Przybylska-Mazur: Selected methods of the determination of

core inflation ... 334

Andrzej Rutkowski: The profitability of acquiring companies listed on the

Warsaw Stock Exchange ... 346

Dorota Skała: Striving towards the mean? Income smoothing dynamics in

small Polish banks ... 364

Piotr Staszkiewicz, Lucia Staszkiewicz: HFT’s potential of investment

companies ... 376

Dorota Szczygieł: Application of three-dimensional copula functions in the

analysis of dependence structure between exchange rates ... 390

Aleksandra Szpulak: A concept of an integrative working capital

manage-ment in line with wealth maximization criterion ... 405

Magdalena Walczak-Gańko: Comparative analysis of exchange traded

products markets in the Czech Republic, Hungary and Poland ... 426

Stanisław Wanat, Monika Papież, Sławomir Śmiech: Causality in

distribu-tion between European stock markets and commodity prices: using inde-pendence test based on the empirical copula ... 439

Krystyna Waszak: The key success factors of investing in shopping malls on

the example of Polish commercial real estate market ... 455

Ewa Widz: Single stock futures quotations as a forecasting tool for stock

prices ... 469

Tadeusz Winkler-Drews: Contrarian strategy risks on the Warsaw Stock

Ex-change ... 483

Marta Wiśniewska: EUR/USD high frequency trading: investment

perfor-mance ... 496

Agnieszka Wojtasiak-Terech: Risk identification and assessment −

guide-lines for public sector in Poland ... 510

Ewa Wycinka: Time to default analysis in personal credit scoring ... 527 Justyna Zabawa, Magdalena Bywalec: Analysis of the financial position

of the banking sector of the European Union member states in the period 2007–2013 ... 537

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Contents

7 Streszczenia

Roman Asyngier: Efekt resplitu na Giełdzie Papierów Wartościowych

w Warszawie ... 25

Monika Banaszewska: Inwestorzy zagraniczni na polskim rynku obligacji

skarbowych w latach 2007–2013 ... 35

Katarzyna Byrka-Kita, Mateusz Czerwiński: Transakcje dotyczące

zna-czących pakietów akcji a prywatne korzyści z tytułu kontroli na polskim rynku kapitałowym ... 49

Ewa Dziwok: Ocena umiejętności inwestycyjnych dla portfela o stałym

do-chodzie ... 58

Łukasz Feldman: Zarządzanie ryzykiem w gospodarstwach domowych

z wykorzystaniem międzyokresowego modelu konsumpcji ... 71

Jerzy Gwizdała: Odwrócony kredyt hipoteczny na wybranych światowych

rynkach kredytów mieszkaniowych ... 85

Magdalena Homa: Rezerwy matematyczne składek UFK a rzeczywista

war-tość portfela referencyjnego ... 97

Monika Kaczała, Dorota Wiśniewska: Zagrożenia w gospodarstwach

rol-nych w Polsce i finansowanie ich skutków – wyniki badań ... 114

Yury Y. Karaleu: Podejście „Slice-Of-Life” do dostosowania modeli

upadło-ściowych na Białorusi ... 131

Patrycja Kowalczyk-Rólczyńska: Produkty typu equity release jako forma

zabezpieczenia emerytalnego ... 140

Dominik Krężołek: Wybrane modele zmienności i ryzyka na przykładzie

rynku metali ... 156

Bożena Kunz: Zakres ujawnianych informacji w ramach metod wyceny

wartości godziwej instrumentów finansowych w sprawozdaniach finanso-wych banków notowanych na GPW ... 175

Szymon Kwiatkowski: Venture debt – instrumenty finansowe i ryzyko

inwe-stycyjne funduszy finansujących wczesną fazę rozwoju przedsiębiorstw .. 184

Katarzyna Łęczycka: Ocena dokładności modelowania zmienności indeksu

VIX z zastosowaniem modelu GARCH ... 198

Ewa Majerowska: Podejmowanie decyzji inwestycyjnych: analiza

technicz-na a modelowanie procesów fitechnicz-nansowych ... 209

Agnieszka Majewska: Formuła ceny wykonania w opcjach menedżerskich –

testowanie proponowanego podejścia ... 221

Sebastian Majewski: Efektywność informacyjna piłkarskiego rynku

bukma-cherskiego w Polsce ... 234

Marta Małecka: Testy gęstości spektralnej w analizie korelacji przekroczeń

VaR ... 249

Adam Marszk: Rynki akcji krajów BRIC: poziom rozwoju i znaczenie

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8

Contents Aleksander R. Mercik: Ryzyko niewypłacalności kontrahenta na rynku

in-strumentów pochodnych ... 274

Josef Novotný: Wykorzystanie analizy psychologicznej w inwestycjach na

rynku akcji ... 288

Krzysztof Piasecki: Dyskontowanie pod wpływem awersji do ryzyka

termi-nu – przypadek czasu dyskretnego ... 298

Aleksandra Pieloch-Babiarz: Inicjacja wypłaty dywidend jako sygnał

przy-szłych dochodów spółek notowanych na warszawskim parkiecie ... 313

Radosław Pietrzyk, Paweł Rokita: Koncepcja modelu optymalizacji planu

finansowego gospodarstwa domowego ... 333

Agnieszka Przybylska-Mazur: Wybrane metody wyznaczania inflacji

bazo-wej ... 345

Andrzej Rutkowski: Rentowność spółek przejmujących notowanych na

Giełdzie Papierów Wartościowych w Warszawie ... 363

Dorota Skała: Wyrównywanie do średniej? Dynamika wygładzania docho-dów w małych polskich bankach ... 375

Piotr Staszkiewicz, Lucia Staszkiewicz: Potencjał handlu algorytmicznego

firm inwestycyjnych ... 389

Dorota Szczygieł: Zastosowanie trójwymiarowych funkcji copula w analizie

zależności między kursami walutowymi ... 404

Aleksandra Szpulak: Koncepcja zintegrowanego zarządzania operacyjnym

kapitałem pracującym w warunkach maksymalizacji bogactwa inwestorów 425

Magdalena Walczak-Gańko: Giełdowe produkty strukturyzowane – analiza

porównawcza rynków w Czechach, Polsce i na Węgrzech ... 438

Stanisław Wanat, Monika Papież, Sławomir Śmiech: Analiza

przyczynowo-ści w rozkładzie między europejskimi rynkami akcji a cenami surowców z wykorzystaniem testu niezależności opartym na kopule empirycznej ... 454

Krystyna Waszak: Czynniki sukcesu inwestycji w centra handlowe na

przy-kładzie polskiego rynku nieruchomości komercyjnych ... 468

Ewa Widz: Notowania kontraktów futures na akcje jako prognoza przyszłych

cen akcji ... 482

Tadeusz Winkler-Drews: Ryzyko strategii contrarian na GPW w

Warsza-wie ... 495

Marta Wiśniewska: EUR/USD transakcje wysokiej częstotliwości: wyniki

inwestycyjne ... 509

Agnieszka Wojtasiak-Terech: Identyfikacja i ocena ryzyka – wytyczne dla

sektora publicznego w Polsce ... 526

Ewa Wycinka: Zastosowanie analizy historii zdarzeń w skoringu kredytów

udzielanych osobom fizycznym ... 536

Justyna Zabawa, Magdalena Bywalec: Analiza sytuacji finansowej sektora

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PRACE NAUKOWE UNIWERSYTETU EKONOMICZNEGO WE WROCŁAWIU nr 207

RESEARCH PAPERS OF WROCŁAW UNIVERSITY OF ECONOMICS nr 381 • 2015

Financial Investment and Insurance – ISSN 1899-3192 Global Trends and the Polish Market e-ISSN 2392-0041

Agnieszka Przybylska-Mazur

University of Economics in Katowice

e-mail: agnieszka.przybylska-mazur@ue.katowice.pl

Summary: Core inflation can be defined as the part of price change, which is used to assess

the medium-and long-run growth trend in the prices of consumer goods and services in the economy. Usually it is assumed that core inflation is linked to expected inflation and de-mand pressure and is independent of supply shocks. The core inflation indicators allow for assessing the inflationary processes in the economy and therefore they are helpful in choos-ing investment and monetary decisions in the medium-run and long-run. We can distchoos-inguish many measures of core inflation. Therefore, the purpose of this article is use alternative measures of core inflation based on selected low pass filter, such as: exponentially weighted moving average, Holt’s exponential smoothing, Hodrick-Prescott filter and Baxter-King fil-ter. In the article we empirically compare these alternative measures with traditional measures of core inflation calculated by the NBP.

Keywords: Core inflation, trimmed mean, exponential smoothing, low-pass filter.

DOI: 10.15611/pn.2015.381.25

1. Introduction

The core inflation illustrates the tendency of price changes adjusted from the tran-sient fluctuations. If the inflation rate is above the core inflation, we can predict that future inflation should decline because the transitional elements that cause high current level of inflation will expire.

Therefore, the core inflation measures allow a more accurate assessment of in-flationary processes in the economy. Moreover, the core inflation measures have often lower variation over time than CPI inflation rates.

There are many measures of core inflation. Currently, the Polish National Bank calculates four core inflation measures: three are computed using mechanical methods, based on the exclusion of a basket of goods from the CPI basket, and one measure is calculated using a statistical method.

The purpose of this paper is the description and application alternative measures of core inflation based on the selected low-pass filters. We compare these

SELECTED METHODS OF THE DETERMINATION

OF CORE INFLATION

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Selected methods of the determination of core inflation

335

alternative measures with traditional measures of core inflation calculated in the Polish National Bank.

When we calculate the traditional measures of core inflation, at the beginning we must exclude certain goods and services from basket used to calculate the CPI inflation rate. Here we have a problem: Price changes of which goods and services are transient? And therefore the exclusion of which goods and services will permit a more accurate assessment of inflationary processes in the economy? However, the presented alternative measures of core inflation, calculated on the basis of low pass filters permit the inclusion of permanent changes in the inflation rate without setting the group of goods or services that should be excluded from the basket, before calculating the core inflation measures. That is necessary for the calculation of the traditional measures of core inflation.

2. Core inflation

The core inflation we define as a part of the headline inflation, which is used to assess the medium and long-term trend of rising prices of consumer goods and services in the economy. Thus, the core inflation

π

bt due to Bryan and Cecchetti

[1994] is as follows: , H t t bt =Eπ + π ₍₁₎

where: πt – the actual inflation rate CPI; H – sufficiently long forecast horizon.

In the next parts we quote traditional measures of core inflation and we present measures of core inflation based on the low-pass filters.

3. Traditional measures of core inflation

T

he Polish National Bank (PNB) calculates the measures of core inflation since 1998. From March 2009 the Polish National Bank has been providing the follow-ing measures of core inflation [NBP 2014]:

1) measures calculated by using mechanical methods, such as: • inflation excluding food and energy prices,

• inflation excluding administered prices, • inflation excluding the most volatile prices,

2) measure calculated by using statistical methods.

Used mechanical methods rely on the elimination of certain elementary price indices from general index of good prices and services and then the calculation of the aggregate price index on a cut off set of goods and services. Most often there are excluded those goods and services whose prices are characterized by strong

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336

Agnieszka Przybylska-Mazur

fluctuations derived from the impact of supply shocks or those goods and services whose information about their changes is limited. The disadvantage of this group of methods is the predetermination of the group of excluded goods.

For example, inflation excluding food and energy prices eliminates the prices of goods that are particularly sensitive to internal and external supply shocks. Thus, this excluded group includes prices of food and non-alcoholic drinks and energy prices. The share of food and energy prices is 42.9% of the CPI basket in 2014.

Inflation rate that is excluding administered prices is aggregate price index cal-culated on the cut off set. It is formed by the elimination of the prices that are not subject to market forces, but are subject to various types of administrative regula-tion. Therefore, the distribution of these prices may not reflect the long-term infla-tion trends Administered prices include goods and services of which final prices are influenced fully or largely by government institutions, local government institu-tions and regulators, and also goods and services whose price changes are legally limited.

The inflation rate excluding most volatile prices is formed by excluding the goods and services whose prices are most disturbed, namely the goods and services whose prices are particularly sensitive to various types of demand and supply shocks or characterized by significant and time-varying seasonality. Strong varia-bility of certain prices can cause the CPI inflation rate does not indicate correctly a long-term trend in the general price level, at least in the short term. Therefore, we calculate the measure of core inflation, which excludes the goods and services whose have the most volatile prices. When the Polish National Bank calculates the measures of core inflation measures, the basket of most variable prices is deter-mined by empirical method and then this basket is deterdeter-mined by using the stand-ard deviation criterion from the sample at the level of elementary price indices. Every year the most variable price basket includes goods and services that are se-lected as long as their total weight achieves 20% of the CPI basket. The goods and services whose prices are the most variable are most often a significant part of foods, mainly unprocessed and low-processed fruits, vegetables, meat and also energy products, such as fuel, gas as well as Internet access services, administra-tion services state and legal services.

There are considered the measures calculated by using statistical methods, such as (α, β) – trimmed mean. This mean is calculated from the formula

, 1 1 ,

∑

+ = − = r m j j x m r xαβ where: 0≤α β, ≤0,5, m=[ ],αn r n= −[ ].βn

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337

When we calculate the trimmed mean we need to order the observations from smallest to largest, and then we reject the (α · 100%) of the smallest values of ob-servation and (β · 100%) of the largest values of obob-servation.

When we calculate this measure of core inflation we often take equal values α and β.

The use of trimmed mean as the measure of core inflation was also proposed by Bryan and Cecchetti [1994].

The Polish National Bank calculates the 15% trimmed mean and it assumes

α = β = 0.15. Then this measure of core inflation is a weighted average calculated

on a set of good prices whose cumulative weights correspond to the previously ascending ordering of price indices that are greater than 15% and less than 85%. The cutting off is done symmetrically on both sides of the distribution of price changes. Thus, we reject the group of goods with a total weight of 30%, which prices have been the biggest change compared to the base period.

The traditional measures of core inflation are less variable than CPI but they re-tain a substantial amount of high frequency variation. Therefore below we present the alternative measures of core inflation received as a result of the low pass filters applied to CPI inflation.

4. Measures based on the low-pass filters

Measures of core inflation based on the filters allow the inclusion of these measures of permanent changes in inflation.

Low-pass filters are one of the types of linear filters. At the beginning we pre-sent some basic notions about filters. The linear filter converts a time series xt

called the input vector into another time series yt called the output vector by the

following linear transformation:

,

∑

∞ −∞ = − ⋅ = k k t k t w x y (2)

where wk are called filter coefficients. We assume that the filter coefficients are

sym-metrical, _k _k

k

w =

−

w

∧

. If we use the linear filters to transformation of variables that are characterized by a trend, then desirable property of the filter is the elimination of this trend. Then the sum of the filter coefficients must be equal to zero,

0 =

∑

∞ −∞ = k wk .

Input vector xt is also called the signal. In order to analyze filters in the

fre-quency domain, we shall consider a signal xt with a known frequency f and xt =

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338

A useful tool for describing and classifying linear filters is the frequency re-sponse function H(f) which, in discrete time setting, is defined by:

, ) (

∑

∞ 2 −∞ = − ⋅ = k k f i k e w f H π ₍₃₎

where: f – frequency; wk – impulse response function.

Therefore, frequency response function is the Fourier transform of the impulse response.

The frequency response function can also be written in the following form:

, ) ( ) (_f _G _f _ei ( f) H ₌ _⋅ ⋅θ (4) where: G(f) – gain function; θ(f) – chase function.

We can now present the definition of low-pass filter. If the value of the gain function is large at low frequencies and small at higher frequencies, the filter is called a low-pass filter. Then the low-frequency dynamics of the inputs are pre-served during the filtering, while the high-frequency components are discarded.

The ideal low-pass filter would have a well-defined cutoff frequency with the frequency response function given by the following formula (see [Gençay, Selçuk, Whitcher 2001]:    < < = otherwise 0 0 for 1 ) (_f f fU H , (5)

where fU is the upper cutoff frequency of the filter, fU < ½.

However, the ideal low-pass filter is not computationally realizable because it requires infinitely many coefficients and an infinite number of data points. Thus, in practical application we use filters containing finitely many coefficients. These filters are only an approximation of the ideal filter.

As filters approximating the ideal low-pass filter we propose to use: 1) exponentially weighted moving average (EWMA),

2) Holt’s exponential smoothing, 3) Hodrick-Prescott filter (HP filter), 4) Baxter-King filter (BK filter), which are presented below.

Moreover, since the calculation of the alternative measures of core inflation used overall CPI inflation, it is sufficient to forecast the core inflation rate based on the low-pass filter the knowledge of CPI or C-CPI forecasts.

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339

4.1. Exponentially Weighted Moving Average (EWMA)

Exponentially Weighted Moving Average EWMA (see: [Zeliaś, Pawełek, Wanat 2003]), called also exponential smoothing we calculate from the following formula:

1 (1 )

ct t c t

π

= ⋅ + −

α π

⋅ ₋ for t > , 1 (6) where: πct – the exponential smoother value, estimate of the average inflation rate,

representing the core inflation rate at the period t; πt – inflation rate at the

period t; α – smoothing constant called also the gain parameter, which is assumed to lie between 0 and 1. Parameter gain is chosen most often a pri-ori. This parameter can be fixed experimentally on the basis of initial sam-ple. Then we calculate the smoothed values for different values α and we choose the value α for which mean error is smallest.

As πc1 we take frequently the value of the initial inflation rate π1 or the

arithme-tic mean of the actual values of the initial sample of real inflation rates (e.g. mean of the first four values).

The rule described by formula (6) can also be written in the following form (see: [Cogley 2002]): 1 [1 (1 ) ] ct L t π _{= ⋅ − −}α α − _{⋅ =}π 0 (1 )k t k k α ∞ α π − = ⋅

∑

− ⋅ ,

where L is lag operator. Thus, formula (6) showing exponential smoothing is a one-sided low-pass filter with exponential weights wk = α(1 – α)k. In practice, when we

calculate the exponential moving average we take into account a finite number of

data values M. Then we often assume = ₊_M

12

α

.

4.2. Holt’s Exponential Smoothing

Equation (6) can be modified and written in the form of Holt’s exponential smooth-ing as followsmooth-ing: 1 1 (1 ) ( ) ct t c t Ct

π

= ⋅ + −

α π

⋅ ₋ + ₋ for t>1, (7) where: 1 for ) 1 ( ) ( − ₁ + − ⋅ ₁ > ⋅ = ₋ C₋ t Ct

β

π

ct

π

ct

β

t (8)

and πct – the smoother value, estimate of the average inflation rate, representing the

core inflation rate at the period t, πt – the inflation rate at period t, α, β – smoothing

constant, α∈(0,1), β∈(0,1), Ct – smoother value of difference in estimated trend

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340

As an initial values πc1 and C1 we can take πc1 = π1 and C1 = y2 – y1 or the

con-stant term and the slope coefficient of the linear trend function, respectively. 4.3. Hodrick-Prescott Filter

Hodrick-Prescot filter (see: [Rubaszek 2012]) we define as the additive decomposi-tion procedure of the time series, in our analysis the CPI infladecomposi-tion rate πt, on the

cyclical component ct and the smoother component representing the core inflation

πct. Therefore

t ct ct

π π

= + , for t = 1, 2, … n. (9)

King and Rebelo [1993] showed that the cyclical component ct of the HP filter

has the following frequency response function (see: [Gençay, Selçuk, Whitcher 2001]): 2 2 4 [1 cos(2 )] ( , ) 1 4 [1 cos(2 )] f H f f λ π λ λ π ⋅ ⋅ − = + ⋅ ⋅ − . (10)

They also showed that the smooth component of the HP filter is given by the following formula (see: [King, Rebelo 1993]):

1 2 1 1 2 2 0 1 1 2 2 0 ( ) ( ) k k ct t k k k k t k k A A A A θ θ λ π θ θ π θ θ π ∞ ⋅ − = ∞ + =  = ⋅_ ⋅ + ⋅ ⋅ +   + ⋅ + ⋅ ⋅ _ 

∑

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where: θ1, θ2 – complex conjugates whose values depend on λ; A1, A2 – the function

of θ1, θ2.

Hence, the filter written in the above form is an infinite order moving average. Therefore, it cannot be implemented in practice without making some assumptions about the lag length.

Thus, in practice, the core inflation component πct is determined using the HP

filter as a solution of the following minimization problem [Gençay, Selçuk, Whitcher 2001]: 1 2 2 1 1 1 2 ( ) [( ) ( )] min n n t ct c t ct ct c t t t π π λ − π + π π π − = = − + − − − →

∑

, (12)

where λ – smoothing parameter, λ ≥ 0.

A problem in the use of the HP filter is appropriate choice of the smoothing pa-rameter. The larger is the value of λ, the less fluctuation is present in the smooth component. When λ = 0 the smooth component is the data itself πct = πt and

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341

smoothing does not occur. When λ → ∞ the smoother component πct resulting from

the application of HP filter is the value of the deterministic linear trend estimated ordinary least squares method.

In application, λ is set to 1600 for quarterly data as suggested in 1997 by Holdrick and Prescott. That provides separation between trend and cycle for about 10 years.

The minimizing problem (12) can be written equivalently in matrix form as fol-lows: ( ) ( )T T ( )T min c c λ c K K c Π − Π ⋅ Π − Π + ⋅Π ⋅ ⋅ ⋅ Π → (13) where: Π =

[

π π

1 2 

π

n

]

, Π = c 

π

c1

π

c2 

π

c n, 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 0 0 0 0 0 0 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 K −    ₋     −    =    ₋    −    ₋                 .

Solving the minimizing problem, that is the vector of core inflation rates we can calculate by the formula:

1

( T)

c In λ K K −

Π = Π⋅ + ⋅ ⋅ . (14)

In the formula (14) In is the identity matrix of order n.

4.4. Baxter-King Filter

Baxter and King (1999) proposed the following estimator of the ideal filter for a finite number of observations:

,

∑

− = − ⋅ = N N k k t k t c w π π (15)

N is a parameter that must be determined before the calculation of the filter

coeffi-cients. For example, Baxter and King suggested that value of N be set at three years. Thus, the Baxter-King filter shows a centered moving average with symmetric weights. If we consider the low-pass filter when the sum of the coefficients wk is

equal to 1, then N _k 1 k N w =− =

∑

.

The optimal values of the filter coefficients wi we calculate from the formulas

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342

Agnieszka Przybylska-Mazur 1 2 1 1 N k k N j j N w v + v =−   = + ⋅ −_ _ 

∑

, (16) where:     = ≠ ⋅ = 0 for 0 for ) sin( 1 k k f k v k _f k π π _,

f – established value of frequency.

Using the low-pass Baxter-King filter the core inflation rates expressed by the formula (15) are calculated on the basis of observations of inflation rates πt for

t = 1, 2, …, n and the value of the core inflation rate can be calculated only at the

periods t = N +1, N + 2, …, n – N.

Then a number of core inflation rates is smaller than number of initial inflation rates. The number of core inflation rates is less about the N initial and N final words.

Therefore, in practical applications to analysis of the current situation or predict future values of the variable, Baxter-King filter is used less frequently than the other filters.

5. Empirical analysis

In our analysis we have into account the monthly CPI inflation in Poland during the period January 2001 – July 2014.

Figure 1. Traditional measures of core inflation

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343

Figure 1 presents the traditional measures of core inflation determined by the Polish National Bank: inflation excluding food and energy prices, inflation exclud-ing the most variable prices, inflation excludexclud-ing administered prices and 15% trimmed mean ((0.15, 0.15) – trimmed mean).

Figures 2a and b present the alternative measures of core inflation calculated on the basis of selected low-pass filters: exponentially weighted moving average, Holt’s exponential smoothing, Hodrick-Prescott filter and Baxter-King filter. We assume gain parameters equal to 0.5 and for the BK filter N = 36.

Figure 2a. Alternative measures of core inflation

Source: own calculations.

Figure 2b. Alternative measures of core inflation (cont)

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344

In order to assess the determined measures of core inflation we calculated the absolute errors – the difference between CPI inflation rate and various measures of core inflation, which are summarized in the figure below.

-2,00,0 2,0 4,0 6,0 8,0 st y-01 st y-02 st y-03 st y-04 st y-05 st y-06 st y-07 st y-08 st y-09 st y-10 st y-11 st y-12 st y-13 st y-14

difference 1 (CPI minus CPI excluding food and energy)

-2,0 -1,0 0,0 1,0 2,0 st y 0 1 st y 0 2 st y 0 3 st y 0 4 st y 0 5 st y 0 6 st y 0 7 st y 0 8 st y 0 9 st y 1 0 st y 1 1 st y 1 2 st y 1 3 st y 1 4

difference 2 (CPI minus CPI excluding the most volatile prices) -2,00,0 2,0 4,0 6,0 8,0

sty sty sty sty sty sty sty sty sty sty sty sty sty sty difference 3 (CPI minus excluding administered prices)

-1,0 -0,5 0,0 0,5 1,0 1,5 2,0 st y 0 1 st y 0 2 st y 0 3 st y 0 4 st y 0 5 st y 0 6 st y 0 7 st y 0 8 st y 0 9 st y 1 0 st y 1 1 st y 1 2 st y 1 3 st y 1 4

difference 4 (CPI minus 15% trimmed mean)

difference 5 (CPI minus EWMA)

difference 6 (CPI minus Holt's exponential smoothing)

-2,0 -1,00,0 1,0 2,0 3,0 4,0 5,0 st y 0 1 st y 0 2 st y 0 3 st y 0 4 st y 0 5 st y 0 6 st y 0 7 st y 0 8 st y 0 9 st y 1 0 st y 1 1 st y 1 2 st y 1 3 st y 1 4

difference 7 (CPI minus HP filter)

-2,0 -1,0 0,0 1,0 2,0 3,0 st y 04 st y 05 st y 06 st y 07 st y 08 st y 09 st y 10 st y 11

difference 8 (CPI minus BK filter)

Figure 3. The absolute errors

Source: own calculations.

We note that throughout the period considered the smallest differences were ob-tained for core inflation calculated by alternative methods: the exponentially weighted moving average, Holt’s exponential smoothing and Hodrick-Prescott

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345

filter. Other results we obtained for the measure of core inflation determined on base the Baxter-King filter.

At the end of the evaluated period, the exact results give also the selected clas-sical measures of core inflation: inflation excluding food and energy prices, infla-tion excluding administered prices and the 15% trimmed mean.

6. Conclusions

For different economic conditions considered measures of core inflation approxi-mate unobservable upward trend in the general price level with different accuracy. The usefulness of these measures largely depends on the nature of shocks which yield price categories excluded from the various core inflation measures.

References

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Eco-nomic Time Series, The Review of EcoEco-nomics and Statistics, vol. 81, no. 4, p. 575–593.

Bryan M.F., Cecchetti S.G., 1994, Measuring Core Inflation, [in:] N.G. Mankiw (ed.), Monetary

Policy, Chicago University of Chicago Press, p. 195–215.

Cogley T., 2002, A Simple Adaptive Measure of Core Inflation, Journal of Money, Credit and Bank-ing”, vol. 34, no. 1, p. 94–113.

Gençay R., Selçuk F., Whitcher B., 2001, An Introduction to Wavelets and Other Filtering Methods

in Finance and Economics, Academic Press, San Diego.

King R.G., Rebelo S.T., 1993, Low Frequency Filtering and Real Business Cycles, Journal of Eco-nomic Dynamic and Control, vol. 17, p. 207–231.

NBP, 2014, Metodyka obliczania miar inflacji bazowej publikowanych przez Narodowy Bank Polski, Instytut Ekonomiczny NBP, Warszawa.

Rubaszek M., 2012, Modelowanie polskiej gospodarki z pakietem R, Oficyna Wydawnicza SGH, Warszawa.

Zeliaś A., Pawełek B., Wanat S., 2003, Prognozowanie ekonomiczne. Teoria przykłady zadania, Wydawnictwo Naukowe PWN, Warszawa.

WYBRANE METODY WYZNACZANIA INFLACJI BAZOWEJ Streszczenie: Inflację bazową można definiować jako część inflacji rejestrowanej, która służy

do oceny średnio- i długookresowego trendu wzrostu cen towarów i usług konsumpcyjnych w gospodarce. Zazwyczaj przyjmuje się, że inflacja bazowa jest związana z oczekiwaniami inflacyjnymi i presją popytową oraz jest niezależna od szoków podażowych. Wskaźniki inflacji bazowej umożliwiają pogłębioną ocenę procesów inflacyjnych w gospodarce, a zatem są pomocne w podejmowaniu decyzji inwestycyjnych i monetarnych w średnim i długim okresie. Można wyróżnić wiele miar inflacji bazowej. W związku z tym w artykule zostały zaprezentowane wybrane metody wyznaczania inflacji bazowej.

Słowa kluczowe: inflacja bazowa, średnia obcięta, wygładzenie wykładnicze, filtr